3.1.4 \(\int \frac {\text {csch}^{-1}(a+b x)}{x} \, dx\) [4]

3.1.4.1 Optimal result

Integrand size = 10, antiderivative size = 162 \[ \int \frac {\text {csch}^{-1}(a+b x)}{x} \, dx=\text {csch}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {csch}^{-1}(a+b x)}}{1-\sqrt {1+a^2}}\right )+\text {csch}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {csch}^{-1}(a+b x)}}{1+\sqrt {1+a^2}}\right )-\text {csch}^{-1}(a+b x) \log \left (1-e^{2 \text {csch}^{-1}(a+b x)}\right )+\operatorname {PolyLog}\left (2,\frac {a e^{\text {csch}^{-1}(a+b x)}}{1-\sqrt {1+a^2}}\right )+\operatorname {PolyLog}\left (2,\frac {a e^{\text {csch}^{-1}(a+b x)}}{1+\sqrt {1+a^2}}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(a+b x)}\right ) \]

output

-arccsch(b*x+a)*ln(1-(1/(b*x+a)+(1+1/(b*x+a)^2)^(1/2))^2)+arccsch(b*x+a)*l 
n(1-a*(1/(b*x+a)+(1+1/(b*x+a)^2)^(1/2))/(1-(a^2+1)^(1/2)))+arccsch(b*x+a)* 
ln(1-a*(1/(b*x+a)+(1+1/(b*x+a)^2)^(1/2))/(1+(a^2+1)^(1/2)))-1/2*polylog(2, 
(1/(b*x+a)+(1+1/(b*x+a)^2)^(1/2))^2)+polylog(2,a*(1/(b*x+a)+(1+1/(b*x+a)^2 
)^(1/2))/(1-(a^2+1)^(1/2)))+polylog(2,a*(1/(b*x+a)+(1+1/(b*x+a)^2)^(1/2))/ 
(1+(a^2+1)^(1/2)))
 

3.1.4.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.48 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.64 \[ \int \frac {\text {csch}^{-1}(a+b x)}{x} \, dx=\frac {1}{8} \left (\pi ^2-4 i \pi \text {csch}^{-1}(a+b x)-8 \text {csch}^{-1}(a+b x)^2-32 \arcsin \left (\frac {\sqrt {\frac {-i+a}{a}}}{\sqrt {2}}\right ) \arctan \left (\frac {(1-i a) \cot \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(a+b x)\right )\right )}{\sqrt {1+a^2}}\right )-8 \text {csch}^{-1}(a+b x) \log \left (1-e^{-2 \text {csch}^{-1}(a+b x)}\right )+4 i \pi \log \left (1-\frac {\left (-1+\sqrt {1+a^2}\right ) e^{\text {csch}^{-1}(a+b x)}}{a}\right )+8 \text {csch}^{-1}(a+b x) \log \left (1-\frac {\left (-1+\sqrt {1+a^2}\right ) e^{\text {csch}^{-1}(a+b x)}}{a}\right )+16 i \arcsin \left (\frac {\sqrt {\frac {-i+a}{a}}}{\sqrt {2}}\right ) \log \left (1-\frac {\left (-1+\sqrt {1+a^2}\right ) e^{\text {csch}^{-1}(a+b x)}}{a}\right )+4 i \pi \log \left (1+\frac {\left (1+\sqrt {1+a^2}\right ) e^{\text {csch}^{-1}(a+b x)}}{a}\right )+8 \text {csch}^{-1}(a+b x) \log \left (1+\frac {\left (1+\sqrt {1+a^2}\right ) e^{\text {csch}^{-1}(a+b x)}}{a}\right )-16 i \arcsin \left (\frac {\sqrt {\frac {-i+a}{a}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (1+\sqrt {1+a^2}\right ) e^{\text {csch}^{-1}(a+b x)}}{a}\right )-4 i \pi \log \left (\frac {b x}{a+b x}\right )+4 \operatorname {PolyLog}\left (2,e^{-2 \text {csch}^{-1}(a+b x)}\right )+8 \operatorname {PolyLog}\left (2,\frac {\left (-1+\sqrt {1+a^2}\right ) e^{\text {csch}^{-1}(a+b x)}}{a}\right )+8 \operatorname {PolyLog}\left (2,-\frac {\left (1+\sqrt {1+a^2}\right ) e^{\text {csch}^{-1}(a+b x)}}{a}\right )\right ) \]

input

Integrate[ArcCsch[a + b*x]/x,x]
 

output

(Pi^2 - (4*I)*Pi*ArcCsch[a + b*x] - 8*ArcCsch[a + b*x]^2 - 32*ArcSin[Sqrt[ 
(-I + a)/a]/Sqrt[2]]*ArcTan[((1 - I*a)*Cot[(Pi + (2*I)*ArcCsch[a + b*x])/4 
])/Sqrt[1 + a^2]] - 8*ArcCsch[a + b*x]*Log[1 - E^(-2*ArcCsch[a + b*x])] + 
(4*I)*Pi*Log[1 - ((-1 + Sqrt[1 + a^2])*E^ArcCsch[a + b*x])/a] + 8*ArcCsch[ 
a + b*x]*Log[1 - ((-1 + Sqrt[1 + a^2])*E^ArcCsch[a + b*x])/a] + (16*I)*Arc 
Sin[Sqrt[(-I + a)/a]/Sqrt[2]]*Log[1 - ((-1 + Sqrt[1 + a^2])*E^ArcCsch[a + 
b*x])/a] + (4*I)*Pi*Log[1 + ((1 + Sqrt[1 + a^2])*E^ArcCsch[a + b*x])/a] + 
8*ArcCsch[a + b*x]*Log[1 + ((1 + Sqrt[1 + a^2])*E^ArcCsch[a + b*x])/a] - ( 
16*I)*ArcSin[Sqrt[(-I + a)/a]/Sqrt[2]]*Log[1 + ((1 + Sqrt[1 + a^2])*E^ArcC 
sch[a + b*x])/a] - (4*I)*Pi*Log[(b*x)/(a + b*x)] + 4*PolyLog[2, E^(-2*ArcC 
sch[a + b*x])] + 8*PolyLog[2, ((-1 + Sqrt[1 + a^2])*E^ArcCsch[a + b*x])/a] 
 + 8*PolyLog[2, -(((1 + Sqrt[1 + a^2])*E^ArcCsch[a + b*x])/a)])/8
 

3.1.4.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.03 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.39, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.600, Rules used = {6876, 25, 6130, 6103, 25, 3042, 26, 4199, 25, 2620, 2715, 2838, 6095, 2620, 2715, 2838}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\text {csch}^{-1}(a+b x)}{x} \, dx\)

\(\Big \downarrow \) 6876

\(\displaystyle -\int \frac {(a+b x)^2 \sqrt {1+\frac {1}{(a+b x)^2}} \text {csch}^{-1}(a+b x)}{b x}d\text {csch}^{-1}(a+b x)\)

\(\Big \downarrow \) 25

\(\displaystyle \int -\frac {(a+b x)^2 \sqrt {\frac {1}{(a+b x)^2}+1} \text {csch}^{-1}(a+b x)}{b x}d\text {csch}^{-1}(a+b x)\)

\(\Big \downarrow \) 6130

\(\displaystyle \int \frac {(a+b x) \sqrt {\frac {1}{(a+b x)^2}+1} \text {csch}^{-1}(a+b x)}{\frac {a}{a+b x}-1}d\text {csch}^{-1}(a+b x)\)

\(\Big \downarrow \) 6103

\(\displaystyle a \int -\frac {\sqrt {1+\frac {1}{(a+b x)^2}} \text {csch}^{-1}(a+b x)}{1-\frac {a}{a+b x}}d\text {csch}^{-1}(a+b x)-\int (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}} \text {csch}^{-1}(a+b x)d\text {csch}^{-1}(a+b x)\)

\(\Big \downarrow \) 25

\(\displaystyle -\int (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}} \text {csch}^{-1}(a+b x)d\text {csch}^{-1}(a+b x)-a \int \frac {\sqrt {1+\frac {1}{(a+b x)^2}} \text {csch}^{-1}(a+b x)}{1-\frac {a}{a+b x}}d\text {csch}^{-1}(a+b x)\)

\(\Big \downarrow \) 3042

\(\displaystyle -a \int \frac {\sqrt {1+\frac {1}{(a+b x)^2}} \text {csch}^{-1}(a+b x)}{1-\frac {a}{a+b x}}d\text {csch}^{-1}(a+b x)-\int -i \text {csch}^{-1}(a+b x) \tan \left (i \text {csch}^{-1}(a+b x)+\frac {\pi }{2}\right )d\text {csch}^{-1}(a+b x)\)

\(\Big \downarrow \) 26

\(\displaystyle -a \int \frac {\sqrt {1+\frac {1}{(a+b x)^2}} \text {csch}^{-1}(a+b x)}{1-\frac {a}{a+b x}}d\text {csch}^{-1}(a+b x)+i \int \text {csch}^{-1}(a+b x) \tan \left (i \text {csch}^{-1}(a+b x)+\frac {\pi }{2}\right )d\text {csch}^{-1}(a+b x)\)

\(\Big \downarrow \) 4199

\(\displaystyle -a \int \frac {\sqrt {1+\frac {1}{(a+b x)^2}} \text {csch}^{-1}(a+b x)}{1-\frac {a}{a+b x}}d\text {csch}^{-1}(a+b x)+i \left (2 i \int -\frac {e^{2 \text {csch}^{-1}(a+b x)} \text {csch}^{-1}(a+b x)}{1-e^{2 \text {csch}^{-1}(a+b x)}}d\text {csch}^{-1}(a+b x)-\frac {1}{2} i \text {csch}^{-1}(a+b x)^2\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -a \int \frac {\sqrt {1+\frac {1}{(a+b x)^2}} \text {csch}^{-1}(a+b x)}{1-\frac {a}{a+b x}}d\text {csch}^{-1}(a+b x)+i \left (-2 i \int \frac {e^{2 \text {csch}^{-1}(a+b x)} \text {csch}^{-1}(a+b x)}{1-e^{2 \text {csch}^{-1}(a+b x)}}d\text {csch}^{-1}(a+b x)-\frac {1}{2} i \text {csch}^{-1}(a+b x)^2\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle -a \int \frac {\sqrt {1+\frac {1}{(a+b x)^2}} \text {csch}^{-1}(a+b x)}{1-\frac {a}{a+b x}}d\text {csch}^{-1}(a+b x)+i \left (-2 i \left (\frac {1}{2} \int \log \left (1-e^{2 \text {csch}^{-1}(a+b x)}\right )d\text {csch}^{-1}(a+b x)-\frac {1}{2} \text {csch}^{-1}(a+b x) \log \left (1-e^{2 \text {csch}^{-1}(a+b x)}\right )\right )-\frac {1}{2} i \text {csch}^{-1}(a+b x)^2\right )\)

\(\Big \downarrow \) 2715

\(\displaystyle -a \int \frac {\sqrt {1+\frac {1}{(a+b x)^2}} \text {csch}^{-1}(a+b x)}{1-\frac {a}{a+b x}}d\text {csch}^{-1}(a+b x)+i \left (-2 i \left (\frac {1}{4} \int e^{-2 \text {csch}^{-1}(a+b x)} \log \left (1-e^{2 \text {csch}^{-1}(a+b x)}\right )de^{2 \text {csch}^{-1}(a+b x)}-\frac {1}{2} \text {csch}^{-1}(a+b x) \log \left (1-e^{2 \text {csch}^{-1}(a+b x)}\right )\right )-\frac {1}{2} i \text {csch}^{-1}(a+b x)^2\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle -a \int \frac {\sqrt {1+\frac {1}{(a+b x)^2}} \text {csch}^{-1}(a+b x)}{1-\frac {a}{a+b x}}d\text {csch}^{-1}(a+b x)+i \left (-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(a+b x)}\right )-\frac {1}{2} \text {csch}^{-1}(a+b x) \log \left (1-e^{2 \text {csch}^{-1}(a+b x)}\right )\right )-\frac {1}{2} i \text {csch}^{-1}(a+b x)^2\right )\)

\(\Big \downarrow \) 6095

\(\displaystyle -a \left (\int \frac {e^{\text {csch}^{-1}(a+b x)} \text {csch}^{-1}(a+b x)}{-e^{\text {csch}^{-1}(a+b x)} a-\sqrt {a^2+1}+1}d\text {csch}^{-1}(a+b x)+\int \frac {e^{\text {csch}^{-1}(a+b x)} \text {csch}^{-1}(a+b x)}{-e^{\text {csch}^{-1}(a+b x)} a+\sqrt {a^2+1}+1}d\text {csch}^{-1}(a+b x)+\frac {\text {csch}^{-1}(a+b x)^2}{2 a}\right )+i \left (-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(a+b x)}\right )-\frac {1}{2} \text {csch}^{-1}(a+b x) \log \left (1-e^{2 \text {csch}^{-1}(a+b x)}\right )\right )-\frac {1}{2} i \text {csch}^{-1}(a+b x)^2\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle -a \left (\frac {\int \log \left (1-\frac {a e^{\text {csch}^{-1}(a+b x)}}{1-\sqrt {a^2+1}}\right )d\text {csch}^{-1}(a+b x)}{a}+\frac {\int \log \left (1-\frac {a e^{\text {csch}^{-1}(a+b x)}}{\sqrt {a^2+1}+1}\right )d\text {csch}^{-1}(a+b x)}{a}-\frac {\text {csch}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {csch}^{-1}(a+b x)}}{1-\sqrt {a^2+1}}\right )}{a}-\frac {\text {csch}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {csch}^{-1}(a+b x)}}{\sqrt {a^2+1}+1}\right )}{a}+\frac {\text {csch}^{-1}(a+b x)^2}{2 a}\right )+i \left (-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(a+b x)}\right )-\frac {1}{2} \text {csch}^{-1}(a+b x) \log \left (1-e^{2 \text {csch}^{-1}(a+b x)}\right )\right )-\frac {1}{2} i \text {csch}^{-1}(a+b x)^2\right )\)

\(\Big \downarrow \) 2715

\(\displaystyle -a \left (\frac {\int e^{-\text {csch}^{-1}(a+b x)} \log \left (1-\frac {a e^{\text {csch}^{-1}(a+b x)}}{1-\sqrt {a^2+1}}\right )de^{\text {csch}^{-1}(a+b x)}}{a}+\frac {\int e^{-\text {csch}^{-1}(a+b x)} \log \left (1-\frac {a e^{\text {csch}^{-1}(a+b x)}}{\sqrt {a^2+1}+1}\right )de^{\text {csch}^{-1}(a+b x)}}{a}-\frac {\text {csch}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {csch}^{-1}(a+b x)}}{1-\sqrt {a^2+1}}\right )}{a}-\frac {\text {csch}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {csch}^{-1}(a+b x)}}{\sqrt {a^2+1}+1}\right )}{a}+\frac {\text {csch}^{-1}(a+b x)^2}{2 a}\right )+i \left (-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(a+b x)}\right )-\frac {1}{2} \text {csch}^{-1}(a+b x) \log \left (1-e^{2 \text {csch}^{-1}(a+b x)}\right )\right )-\frac {1}{2} i \text {csch}^{-1}(a+b x)^2\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle -a \left (-\frac {\operatorname {PolyLog}\left (2,\frac {a e^{\text {csch}^{-1}(a+b x)}}{1-\sqrt {a^2+1}}\right )}{a}-\frac {\operatorname {PolyLog}\left (2,\frac {a e^{\text {csch}^{-1}(a+b x)}}{\sqrt {a^2+1}+1}\right )}{a}-\frac {\text {csch}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {csch}^{-1}(a+b x)}}{1-\sqrt {a^2+1}}\right )}{a}-\frac {\text {csch}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {csch}^{-1}(a+b x)}}{\sqrt {a^2+1}+1}\right )}{a}+\frac {\text {csch}^{-1}(a+b x)^2}{2 a}\right )+i \left (-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(a+b x)}\right )-\frac {1}{2} \text {csch}^{-1}(a+b x) \log \left (1-e^{2 \text {csch}^{-1}(a+b x)}\right )\right )-\frac {1}{2} i \text {csch}^{-1}(a+b x)^2\right )\)

input

Int[ArcCsch[a + b*x]/x,x]
 

output

-(a*(ArcCsch[a + b*x]^2/(2*a) - (ArcCsch[a + b*x]*Log[1 - (a*E^ArcCsch[a + 
 b*x])/(1 - Sqrt[1 + a^2])])/a - (ArcCsch[a + b*x]*Log[1 - (a*E^ArcCsch[a 
+ b*x])/(1 + Sqrt[1 + a^2])])/a - PolyLog[2, (a*E^ArcCsch[a + b*x])/(1 - S 
qrt[1 + a^2])]/a - PolyLog[2, (a*E^ArcCsch[a + b*x])/(1 + Sqrt[1 + a^2])]/ 
a)) + I*((-1/2*I)*ArcCsch[a + b*x]^2 - (2*I)*(-1/2*(ArcCsch[a + b*x]*Log[1 
 - E^(2*ArcCsch[a + b*x])]) - PolyLog[2, E^(2*ArcCsch[a + b*x])]/4))
 

3.1.4.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4199

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_ 
.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp 
[2*I   Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x 
))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In 
tegerQ[4*k] && IGtQ[m, 0]
 

rule 6095

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin 
h[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), 
 x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]
 

rule 6103

Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_ 
.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/a   Int[(e + f*x)^m*Coth[ 
c + d*x]^n, x], x] - Simp[b/a   Int[(e + f*x)^m*Cosh[c + d*x]*(Coth[c + d*x 
]^(n - 1)/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && 
 IGtQ[m, 0] && IGtQ[n, 0]
 

rule 6130

Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.)*(G_)[(c_.) + 
 (d_.)*(x_)]^(p_.))/(Csch[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> I 
nt[(e + f*x)^m*Sinh[c + d*x]*F[c + d*x]^n*(G[c + d*x]^p/(b + a*Sinh[c + d*x 
])), x] /; FreeQ[{a, b, c, d, e, f}, x] && HyperbolicQ[F] && HyperbolicQ[G] 
 && IntegersQ[m, n, p]
 

rule 6876

Int[((a_.) + ArcCsch[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[-(d^(m + 1))^(-1)   Subst[Int[(a + b*x)^p*Csch[x]*C 
oth[x]*(d*e - c*f + f*Csch[x])^m, x], x, ArcCsch[c + d*x]], x] /; FreeQ[{a, 
 b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]
 

3.1.4.4 Maple [F]

\[\int \frac {\operatorname {arccsch}\left (b x +a \right )}{x}d x\]

input

int(arccsch(b*x+a)/x,x)
 

output

int(arccsch(b*x+a)/x,x)
 

3.1.4.5 Fricas [F]

\[ \int \frac {\text {csch}^{-1}(a+b x)}{x} \, dx=\int { \frac {\operatorname {arcsch}\left (b x + a\right )}{x} \,d x } \]

input

integrate(arccsch(b*x+a)/x,x, algorithm="fricas")
 

output

integral(arccsch(b*x + a)/x, x)
 

3.1.4.6 Sympy [F]

\[ \int \frac {\text {csch}^{-1}(a+b x)}{x} \, dx=\int \frac {\operatorname {acsch}{\left (a + b x \right )}}{x}\, dx \]

input

integrate(acsch(b*x+a)/x,x)
 

output

Integral(acsch(a + b*x)/x, x)
 

3.1.4.7 Maxima [F]

\[ \int \frac {\text {csch}^{-1}(a+b x)}{x} \, dx=\int { \frac {\operatorname {arcsch}\left (b x + a\right )}{x} \,d x } \]

input

integrate(arccsch(b*x+a)/x,x, algorithm="maxima")
 

output

integrate(arccsch(b*x + a)/x, x)
 

3.1.4.8 Giac [F]

\[ \int \frac {\text {csch}^{-1}(a+b x)}{x} \, dx=\int { \frac {\operatorname {arcsch}\left (b x + a\right )}{x} \,d x } \]

input

integrate(arccsch(b*x+a)/x,x, algorithm="giac")
 

output

integrate(arccsch(b*x + a)/x, x)
 

3.1.4.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\text {csch}^{-1}(a+b x)}{x} \, dx=\int \frac {\mathrm {asinh}\left (\frac {1}{a+b\,x}\right )}{x} \,d x \]

input

int(asinh(1/(a + b*x))/x,x)
 

output

int(asinh(1/(a + b*x))/x, x)